Where $B_x$ denotes $\partial B / \partial x$ and so on. One way that makes it easier to memorize the formula is to associate the first column and row with $x$, second column and row with $y$, third with $z$, and last with $w$.

Spin

If $G : \mathbb{R}^4 \to S$ is a differentiable function that takes a point in $\mathbb{R}^4$ to a skew-symmetric matrix, we define the operation "spin" that produces a vector field in $\mathbb{R}^4$ as follows:

If $G(x,y,z,w) = \{ \sum_{i,j} G_{ij}(x,y,z,w) \}$ (where $G_{ij}$ is the entry at the $i$-th row and $j$-th column, and since $G$ is skew-symmetric, then $G_{i,i} = 0$ and $G_{ji} = -G_{ij}$)

Then

$(\bold{spin} G)(x,y,z,w) = (G_{23_w} + G_{34_y} + G_{42_z}) \vec i $

$+(G_{13_w} + G_{34_x} + G_{41_z}) \vec j$

$+(G_{12_w} + G_{24_x} + G_{41_y}) \vec k$

$+(G_{12_z} + G_{23_x} + G_{31_y}) \vec l$

where again $G_{ij_x}$ denotes $\partial G_{ij} / \partial x$ and so on.